Liquid dispenser systems, such as liquid soap and sanitizer dispensers, provide a user with a predetermined amount of liquid upon actuation of the dispenser. In addition, it is sometimes desirable to dispense the liquid in the form of foam by, for example, injecting air into the liquid to create a foamy mixture of liquid and air bubbles. Many dispensers are refillable with refill units that comprise a pump (or a pump and an air compressor) and a container. Many of the refill units currently on the market are inverted. In an inverted refill unit, the pump is located under the container when installed in the dispenser and the containers are often “collapsible” containers. That is, as liquid is pumped out of the container, a vacuum is created in the container and the vacuum pressure collapses the container because air does not flow into the container to replace the liquid. Providing a vent in the bottom of the container subjects the container to potential leakage problems.